9 If-Then Statements:

Relations Involving Addition, Subtraction, Multiplication, Division, and Equality

  • This chapter does not provide a model activity for elementary students
  • Discuss basic properties that underlie arithmetic and beginning algebra
  • Together with previously discussed topics this chapter explores the basis for most of arithmetic and beginning algebra

Relating Subtraction to Addition and Division to Multiplication

Addition and Subtraction

  • 9 boys and 12 girls, how many children in the class?
  • 21 children, 9 are boys, how many girls are in the class?
  • Both problems describe the same situation
  • Different problems can be made from this information
  • Different problems require different methods for solving
  • Addition
  • Subtraction
  • Missing addend
  • Demonstrate the relation between addition and subtraction
  • Addition or subtraction can be thought of as positive numbers where a whole is composed of two parts for the set of whole numbers
  • This is commonly thought of as the missing addend model for subtraction
  • Addend + addend = sum for addition of whole numbers
  • Addend + missing addend = sum for subtraction of whole numbers (Sum – addend = missing addend)

The Relation between Addition and Subtraction

  • If a – b = c, then a = c + b
  • If d + e = f, then d = f – e and e = f – d
  • Might use this relation in several commonly encountered contexts
  • Number facts
  • Fact families
  • Can make learning facts easier and more robust
  • Solving word problems
  • Open number sentences
  • Provides for alternative solutions to problems
  • Provides context for discussing the relation between addition and subtraction

Teacher Commentary 9.1

  • 3rd grade students
  • Some subtracted to find answer, but realized addition could also be used
  • Wrote a conjecture to see if subtraction would always work for this type of problem
  • Students encouraged to find big ideas in their conjectures
  • Teacher provided necessary mathematical language to help students focus thinking
  • Students came up with generalized statement for this conjecture using symbols
  • If  +  = , then  -  = 

Multiplication and Division

  • Related in essentially as addition and subtraction
  • Commonly referred to in division as the missing factor model
  • Factor x factor = product for multiplication of whole numbers
  • Factor x missing factor = product for division of whole numbers (Product  factor = missing factor)

The Relation between Multiplication and Division

  • If a  b = c, then a = c x b
  • If d x e = f, then d = f  e for e  0, and e = f  d for d  0
  • Critical for learning division number facts

Operating on Both Sides of the Equal Sign

  • 3rd grade students using relational thinking
  • 345 + 576 = 342 + 574 + d

Operating on Both Sides of the Equal Sign

  • If a = b, then a + c = b + c
  • If a = b, then a x c = b x c
  • If a = b, then a - c = b - c
  • If a = b, then a  c = b  c, c  0
  • Play a central role in solving algebraic equations
  • See 9.1 p. 127

Proving Conjectures about Subtraction and Division

  • Prove by relating to corresponding conjectures for addition and multiplication
  • Prove d – d = 0 for all numbers
  • Prove p  1 = p for all numbers
  • Show why we cannot divide by zero: If r  0 = , then r =  x 0. Thus r could only be zero, not any number
  • Show why we cannot divide zero by zero: If 0  0 = , then 0 =  x 0. Thus any number could replace , which contradicts the Fundamental Theorem of Arithmetic which says the prime factorization of a number is unique
  • Can also consider fractions or the set of rational numbers for each of these ideas

Inverses

  • For every number r except zero, there is a unique number such that
  • Multiplicative inverseof r or the reciprocal of r
  • Any division problem can be recast as a multiplication problem by multiplying by the inverse
  • Any subtraction problem can be recast as an addition problem by adding the opposite
  • Important for students to distinguish vocabulary between inverse and opposite
  • Addition and multiplication have simpler rules than subtraction and division
  • Does not mean we do away with subtraction and division, just want to understand the relation between them

The Basic Properties Revisited

  • Some properties more critical than others
  • Some properties derived from others
  • See table 9.1 p. 130
  • Properties of subtraction and division are missing – WHY?
  • Zero property of multiplication not included – WHY?